The Structure of a Lie Algebra Attached to a Unit Form 
Received:November 04, 2017 Revised:September 01, 2018 
Key Word:
Nakayama algebras finite dimensional simple Lie algebras RingelHall Lie algebras

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571360) and the Natural Science Foundation of Fujian Province (Grant Nos.2016J01006; JZ160427). 

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Abstract: 
Let $n\geq 4$. The complex Lie algebra, which is attached to the unit form $\mathfrak{q}(x_1,x_2,\ldots, x_n)=\sum_{i=1}^nx_i^2(\sum_{i=1}^{n1}x_ix_{i+1})+x_1x_n $ and defined by generators and generalized Serre relations, is proved to be a finitedimensional simple Lie algebra of type $\mathbb{D}_n$, and realized by the RingelHall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra. 
Citation: 
DOI:10.3770/j.issn:20952651.2019.05.004 
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